ABSTRACT Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.

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关于 Schlömilch 级数的总结

摘要 Schlömilch 级数以德国数学家 Oscar Xavier Schlömilch 的名字命名，他于 1857 年根据第一类贝塞尔函数将其作为傅立叶级数展开式导出。然而，除了贝塞尔函数，这里我们也考虑根据 Struve 函数或 Bessel 和 Struve 积分的展开。根据 Bessel 或 Struve 函数获得 Schlömilch 级数和的方法是基于三角级数的和，可以用黎曼 zeta 和倒幂的相关函数表示，在某些情况下可以引入封闭形式，意味着无限级数由有限和表示。通过使用 Krylov 的方法，我们获得了三角级数的收敛加速度。